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Space And Geometry – Part 1

October 22, 2009

Just the other day I was out with my friend Greg, and he was telling me I should write down some of my ideas toward space and time.  After thinking about it for a while, I decided to do so.  This is a subject I’ve been very passionate about for many many years, but rarely talk about it due to the complexity of it all.  When I write blog entries for this site, I tend to write about ideas I can express quickly.   Unfortunately, space and time are very complex ideas, and require a lot of time and effort to explain.

I wrote one entry a while back called “Moving Faster Than Light”, but that was mostly me rambling about the various problems you encounter if a body is allowed to move faster than light.  But in this new series of entries I will take more time and fully explain everything.

Considering the depth of the subject, and my own time constraints, we’ll have to cover our ground step by step.  But if we break this massive subject into enough pieces, it should be manageable – both to read, and for me to write. We’ll start from the very beginning, assuming you know nothing about anything, and move slowly.  My only purpose of this entry today is to explain why we think this world is three dimensional and how we come to that conclusion.   Later we’ll move into things like relativity, quantum mechanics, curved space-time, and more.  For now, we’ll start at the very beginning.

So let’s begin, shall we?  If I was to ask you, “What is space?”, what would you tell me?  How would you explain it to me?  Can it be explained?

If you’re like most people, you probably envision space as a giant box.  Its edges extend out to infinity in all three dimensions.  If I had a rocketship and shot out into space, and kept going in a straight line, I’d just keep going and going and going.  There is no end to it.  I would get farther and farther and farther away from the Earth.

Secondly, you probably think space is “continuous”.  If I was to say, drop a baseball, and you watch it fall to the ground, that motion was smooth and perfect.  There’s no choppiness.  No jerks.  The baseball has a set size, and is a set sphere like shape.  It moves at an absolute speed through the big “space” box, as it accelerates toward the Earth due to the pull of gravity.   In other words, if I was to watch the ball fall with a high speed camera, no matter how quickly it took pictures, there would always be a “frame” for the camera to take.   Even if the camera could take 100 billion pictures a second, and we view the film reel frame by frame, every one of them would be filled with a picture of the baseball, each in a unique position, very very slowly making its way toward the Earth.

Are these assumptions true?  No, not really.  Space is not a giant box of infinite extent.  Motion is not smooth and continuous.  The baseball doesn’t have an absolute size, shape, or velocity.  Its color isn’t even necessarily white, and its seems are not necessarily red.   The Earth doesn’t necessarily orbit the Sun.  It’s just fine to keep your frame of reference on the Earth, and if you do that, the Sun orbits the Earth.  It really can go either way.  And as for the camera and its film reel, if you took enough pictures, you’d eventually find frames which are blank.  In fact, if you took pictures quickly enough, each frame would have at most one pixel lit up and the rest would be blank.  This is because light moves through “space” in energy packets, which are akin to bullets.  The laws behind how these bullets fly gets us into quantum mechanics, but we’ll have to postpone that discussion to a later time.

So are the common assumptions completely wrong?  No.  They’re approximations, and as long as you don’t move at speeds near that of light, don’t extend your 3D box too far, and don’t focus too hard on following each little light photon bullet, you’d be roughly correct.  You’ll be able to do physics and calculate things, and your answers will be correct enough.  And anyways, everything we do in science is an approximation.  There’s no such thing as absolute precision.

So if these ideas are not correct, then how come they’re so common?  Why do we think the world is a 3D box when it isn’t?  Why do we think the baseball has a set size and shape?  Why do we think the room we’re in has set dimensions?  Is it some sort of defect in the brain?  Is the human brain confined to three dimensions, and forced to struggle to understand anything of higher dimensions, such as four dimensional space-time?

No.  The brain could easily understand a four dimensional world.  The problem is not with our brains.  Our problem is we’ve never experienced moving at high speeds near that of light.  That’s not a common experience for us.  Therefore the consequences we draw from Einstein’s equations are strange and foreign to us.  We’ve never seen the baseball morph into an oval, or the straight walls of our house begin to curve then become straight again.  And since our brain works with the experiences it’s had, telling it that, “Yeah, your bedroom wall right there can curve on you if you move fast enough…”, it naturally replies, “… I’ve never seen the wall curve.  That’s a pretty crazy thing to say.  I don’t believe it.”

So let’s talk about how the brain comes to a notion of “space” to begin with, and what that even means.

Space And Your Brain

Tactile And Motor Space

First we have tactile space.  This is a sense of “space” which comes from touching and feeling things. For example, imagine being on the beach with someone you love.  I’ll do the same.  I find myself in an intimate embrace with this lovely woman.  I caress her, my hands moving up and down her body.  I feel every curve.  Even if my eyes are closed, I’m immersed in a very spacious world.

I could be blind my entire life, but I would still have this sense of space.  I could rub my hands across my bed, a bookshelf, a coffee mug, a kitchen chair, or anything of that nature.  Just by rubbing and feeling the object, and the muscular contractions and sensations of touch, I can form a sense of space and of solid objects.

In a very complex way your brain associates various sensations of touch, as well as the the feelings of muscles contracting and loosening, and links them together.

Now the first thing to note is that these sensations and linkages do not necessarily have to include any notions of space whatsoever.  In our day to day experience, most sensations of “touch” occur when our body comes in contact with another body.  But that need not be the case.  It’s just as logical that touch sensations could arise in different circumstances.  For a weird example, imagine if you felt a pressure come over your body the faster you moved relative to some star in space someplace.  The higher your velocity as calculated from someone watching you from that location, the higher the pressure.  It’s possible.  But that’s not how it works in our reality.  We “feel” when we come in contact with something, or a force is acting on our “body” in some way.   The sensations we experience just happen to correspond in such a way lead to our conception of tactile space.

As for the tactile space of everyday experience, it has one interesting property which is worth noting.  Unlike the objects we see with our eyes, those we feel never change size.  To feel something you have to be touching it.  If I embrace this lovely woman I’m with, caress her, then let go, and she takes a few steps away from me out of reach, my sense of touch goes away.  I extend my arms but she’s not there.  As she’s walking away her size may shrink smaller and smaller, but my sense of touch has no parallel to that.  Touch always feels the same in everyday experience and never “shrinks”.

Space Based Upon Sight

Imagine staring at a computer screen which shows random colors.  Kind of like the black and white fuzz on an old television set.  Just random colors appearing all over the screen.

Is there any sense of space there?  No.  But there is a sensation of sight.  You do see colors.  Those colors don’t seem to tie together to make a spacial reality of any sort, but you are seeing something.

So our first point to make is that seeing doesn’t necessarily mean there’s space, or even any “objects” to see.  Objects only come into play when the images change in such a way that there seems to be an order of some sort.  But what do I mean by “order”?

Your first thought may be to think of light waves radiating from the sun, or a light bulb, which then come in contact with the surface of objects, causing their surface atoms to vibrate, which then creates electromagnetic waves of their own, which then radiate back through space, eventually coming in contact with the eyes of various observers, giving rise to a personal idea of space and order.  You further imagine “order” to mean complex laws of physics, which can be patterened with mathematics and logic.   But let’s not get ahead of ourselves.  That may be how the reality we live in right now works, but that’s not to say we couldn’t have a reality with space which works on entirely different principles.

So instead of talking about how alternate realities could work, and give rise to space, let’s discuss how the brain comes to the notions of solid objects, and from those and their various perspectives, gives rise to the 3D geometrical “box” of space that we are all so familiar with.  That’d probably be the most practical, and beneficial concept to understand.

So you’ve just been born.  You’re in your mothers arms and your father is standing at the bedside saying, “You did well”, and “Hey there little guy.”    How did baby Jason come to his first notions of space?  It required three separate concepts.  My visual senses.  Tactile senses.  And Motor senses.

Visual is my ability to see.  Tactile is my ability to feel.  Motor is my ability to move based on my own volition.

I’m not too keen on babies and their early eyesight, but if I recall correctly, babies take a bit of time before they can see well.  Images are initially blurry.  So let’s skip ahead and move on to baby Jason being able to crawl, and capable of seeing clearly.

Mom lays me down on the living room floor, and I begin to squirm around.  What’s going on here?  This is me establishing tactile space.  I feel the hard ground beneath me, I feel my body coming in contact with it, and I also begin to sense how my own volition controls these appendages of mine.  I kick my legs.  Swing my arms.  Clasp various things with my hands.  Stick things in my mouth.  Bite down on everything I can.  And so on.

Eventually I get a feel for my own body, and how it relates with the immediate environment I’m in.  I also notice that as I’m squirming about, the images I see with my eyes keep rotating and spinning.  Eventually though I start to correlate it.

For example, when I move my head, I watch the environment “turn”.  This movement brought about a muscular sensation.  Then I move my head a different way, and I watch the picture I was seeing return back to its initial state.  When I turned my head one way, the picture changed.  But once I moved my head back, the image returned back to how I first saw it.  This is the very beginning of the 3D box space we all know.

Eventually we start to see space using our eyes.  This is because we link our tactile sensations and our visual sensations together, based on our motor movements.  Say there’s a flat screen television mounted on the wall of this living room.  We “aim” our head at the television, then crawl toward it.  We watch the visual sensations change.  But then we crawl backwards, and it returns to the same image from before.  We begin to link our movements with what we see.  Various movements bring us the same images, and tactile sensations, and we correlate those together.  We have formed our first concepts of “location”.

We start to give “objects” a “location”.  An “object” is a set of visual images and tactile feelings, and “location” is us remembering motor movements and subsequent tactile and visual impressions we’ll experience “getting” to that location.  We link everything together in this web of chains of experiences.

So how the images and other sensations change based on our motor volitions, determines space.

To summarize, we see that we could have no concept of space without our sense of touch and our ability to achieve mobility.  We have to be able to move in order to experience different locations.  If we were confined to a single location, and could only see a series of changing visual images, we could never form any conception of space or objects at all.   It would be no different than a slide show of random colors, which would never make any sense to us whatsoever.

Taking in isolation, none of our individual sensations, whether tactile, or visual, could lead us to an idea of “space”.  We only come to think of “space” when we begin to study how these sensations change based on our own motor volition.  We watch their succession, and begin to get a “feel” for space before too long.

Changes Of State and Changes Of Position

So far, the world we’ve described does not allow for things to move, or change state.  In the real world, things move all the time, and objects change in all sorts of ways.  How does our brain come to understand these things?

First let’s discuss position, and an object moving through space.  Let’s go back to the beach, with my beautiful lover.  She kisses me, smiles, then takes a few steps back from me.  Then she takes a few more steps back, and stops.  I’m well aware of her movement.  But what does that mean?  What does it mean for her to “move”, and how did I know about it?

I understand movement in terms of my own movement.  When we kissed, her face was next to my own.  Then she took a few steps back.  In order for me to restore that same visual image of her face, I take a few steps toward her.  I once again have restored the same picture from before.  I see her face up close once again.

Now how do I know whether she moved, or whether I moved?  How do I judge this?  If I see the images change, but feel no accompanying muscular contractions in my own body, I assume that she moved.  I myself have not moved.  If, on the other hand, I feel the muscular contractions, I know that I was the one who brought the change.

So we have a few two scenarios when we’re experiencing a change in sensory impressions:

1.  If I attempted no motor “movement”, and felt no muscular sensations, yet the images and sensations are changing, the object I’m seeing is “moving”.

2. If I attempted motor movement, and feel the muscular sensations, this happens when the object is not moving.  Its stationary and I’m the one bringing about the “movement”.

Now let’s give another example, then discuss what it means for an object to change its state.

My lover is several yards away from me staring me in the eyes.  She apparently likes this little game we’re playing.  She smiles, then she “turns”.  How did my brain recognize this as a turn?  This is because I can “correct” her turn, by walking in a counter circle, relative to how she turned.  I take a few steps in this circle, then once again we’re making eye contact again.

But what about changes in state?  Say a strong breeze blows in, and her long hair is blown into the air.  How did I recognize this as a state change in this lovely woman?  This is due to the fact that I cannot “correct” this change by any motor movements of my own.  I can continue to walk around her in circles, but her hair, which was once hanging down, is now blowing in the wind.  There’s no movement I can do to change that, without going right up to her, and pressing her hair down with my hands.

Watching these state changes is how we come to think of things in “pieces”.  When I saw my lover’s hair blow in the wind, I noticed that her facial position stayed in the same “location” as before.  Her arms, body, and legs, are all in the same “location” as before.  (Remember “location” is motor movements I would have to bring forth to experience those things in a certain way)  Thing is, her hair no longer is in the same location.  It changed location, and I didn’t feel any muscular sensations, so therefore I was not behind the change I’m seeing.

We Finally Define Space!

So what is “space”, then?  Space is our ability to change our impressions through volition, then reestablish those same experiences, and have them once again in relatively the same manner.  Location is a series of motions we can perform which leads to a particular experience we remember, or have been told about, or have possibly imagined.

For example, the flat-panel TV mounted on the wall in the living room.  We’re in the kitchen.  We get out of the chair, walk down the hallway, through the doorway, to the center of the room, turn, and there is the television on the wall, just as we remembered it.  Every time we look at that wall, there the TV is, on the same wall.  We have nearly the same experience each time we walk down the hallway, walk into the living room, turn, and look in that direction.  And if something has changed, because we have a memory, we can say, “Hey, what is this box in the middle of the walkway?  Who left this here?”

This definition of “space” works not just for our own world, but for other “worlds” as well.  Say you’re playing Super Mario on the Nintendo, telling your nerdy girlfriend about a secret passage.  She says, “Where is it?  How do I get there?”  So you both sit down in the living room.  You take the controller into your hands and hit the Nintendo’s power button.  You begin to push buttons on the controller, which brings about certain changes on the TV screen, which you remember.  You move the little Mario on the screen to the pipe, you tell him to go down the pipe, you watch the little Mario guy fall, then land on the ground.  Then you move him some more, and then to the secret passage.  You recognize the graphics which display on the screen and say, “Here it is!”  She then says, “Oh my God I love you!”  Then passionately makes love to you on the living room floor.   Nerdy girls are the best!

Another example is knowing the “location” of a website on the internet.  Location is the perfect word really.  You type in an address into the bar on the web browser and a series of familiar images displays on the screen.  Same thing.  This web address is just as spatial as the “real” location “Paris, France”.  It’s an experience you can bring about if you perform certain actions.   And just as you can get to the same website from multiple computers, you can get to the same location in “actual” space by multiple paths.  It seems almost unconsciously that the people who used the word “location” for a web “address” understood this conception of space.

Controlling Mario on the Nintendo is no different than controlling your real body.  Through a series of voluntary movements you bring about changes in your experiences.  That’s all that space is, in its most general sense.  You’re just as alive playing a video game as you are walking around in the “real” world.  I’m not sure what someone means when they say gamers have “no life.”  When do you stop living?  Life’s just a series of experiences strung together by the twine of our memory.  You can never leave reality.  You’re always there.  You just choose one experience over another.

3D Geometric “Box” Space

Finally we come to the three dimensional conception of space we’re all so familiar with.  It arises as we come to form approximate conceptions of what we’ll call a “solid body”.  In general, a solid body is one where displacements it undergoes can be corrected by our own subsequent movements.

Solid bodies have a tendency to hold their form for long periods time –  long enough for us to unite them together in memory, and form conceptions of space.   If that consistency was not there, we could never form a concept of space.

If everything was like a liquid, constantly flowing this way and that, and never holding to any particular shape or pattern, we could not form a conception of space.  Things have to remain constant.  Laws and patterns must exist.  Geometry and space rely on matter holding to various forms long enough for us to relate them together.

The 3D space we’re acquainted with is formed in our minds due to these solid bodies which exist in our world.  My computer desk is a solid body.  My Playstation 2.  My history book here on the desk.  My computer monitor.  They hold their form and continue to exist for a long period of time.   Throughout my typing of this entire entry, they’ve stayed here on the desk, in their same relative positions, appearing in exactly the same way.

When I form a concept of them, my mind wants to represent them in a three dimensional geometry.   Like my desk for example.  It’s rectangular.  Is its absolute shape rectangular?  No.  On a microscopic level its surface is rough and bumpy.  It’s also mostly empty space, even though to me it appears solid. And as we mentioned earlier, its edges would curve and bend, if I was to approach near light speeds.

But I never move anywhere near light speed.  As I move toward it and away from it at very slow speeds (relative to the speed of light), I see it in a perspective, and all its edges remain straight edged.  It gets smaller as I go far away, and gets larger as I approach it.  The desk’s edges shift at angles based on how I rotate my head, and where I’m standing relative to the desk.

The reason I “see” the desk is because of light coming from the bulb on my ceiling.  Electromagnetic radiation emits from the light on my ceiling, approaches the desk, excites the atoms of the desk’s surface, they radiate light waves back to my eyes, and I see the desk.

I can also feel the desk with my hands, and form a conception of space based on my tactile senses.  With that there is a complex dynamic of forces and atoms, as my hand approaches the desk and comes into contact with it.  I push on the desk, and it pushes back on me with equal and opposite force.  We’ll discuss more of the physics of this phenomenon later.

So by now you should have an idea what space is, and how we form our first ideas of it.  This brings us near the end of this first “lecture” on Space and 3D Geometry.  But there’s much much more to discuss, as we will soon see.  For now, let’s briefly highlight what to expect in our next lecture.

Relativity and Space-Time!

In our next lecture we will discuss how light works, and some of the strange dynamics involved with it.  In particular, we’ll be covering the famous Michelson and Morley experiment.   We’ll see that light is very strange indeed!   Light doesn’t work how you think it does.   Since one of our primary conceptions of space comes from sight, which is based on light, we’ll start to learn some strange things about our world when we look into Einstein’s research.  Also, in later lectures, we’ll discuss the quantum mechanical nature of matter, and attempt to discuss what tactile sensation would be like when moving at near light speeds.  Eventually we’ll find ourselves talking about the big-bang, curved space-time, and the universe.  So look forward to it!

Topics: Philosophy, Physics, Psychology | 1 Comment »

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